Two-dimensional Dirac half-metal ferromagnets and ferromagnetic materials for spintronic devices

ABSTRACT

Ferromagnetic materials are disclosed that comprise at least one Dirac half metal material. In addition, Dirac half metal materials are disclosed, wherein the material comprises a plurality of massless Dirac electrons. In addition, ferromagnetic materials are disclosed that includes at least one Dirac half metal material, wherein the material comprises a plurality of massless Dirac electrons, wherein the material exhibits 100% spin polarization, and wherein the plurality of electrons exhibit ultrahigh mobility. Spintronic devices and heterostructures are also disclosed that include a Dirac half metal material.

This United States Utility Application claims priority to U.S.Provisional Application Ser. No. 62/637,478 filed on Mar. 2, 2018, whichis entitled “Two-Dimensional Dirac Half-Metal Ferromagnets forSpintronic Devices” and which is incorporated herein in its entirety byreference.

The work is supported by NSF-Partnership in Research and Education inMaterials (PREM) Grant DMR-1205734 and NSF Grant No. ERC TANMS-116050.

FIELD OF THE SUBJECT MATTER

The field of the subject matter is Dirac half-metal materials andferromagnets and their use with and in relation to spintronic devices.

BACKGROUND

Spintronics, involving transmission and storage of information bymanipulating the spin degrees of freedom, has sparked tremendousinterest over the past decades, because it offers unique advantages toconventional charge-based electronic devices, such as greater dataprocessing speed, high integration density, low power consumption, andnonvolatility [1]. Several key properties have been identified fordeveloping new magnetic materials for spintronic devices:Room-temperature half-metallicity, large magnetocrystalline anisotropy(MCA), high Curie temperature, and high spin mobility [2, 3].

Bulk half-metals (HMs), with one spin channel conducting and the othersemiconducting are ideal spintronic materials, which exhibit 100% spinpolarization [4-6]. However, in order to preserve the half-metallicityat room temperature the band gap of one spin channel should be wideenough to prevent thermally induced spin-flip transitions. An additionalchallenge is to sustain the half-metallic character in ultrathin HMfilms [7].

Another distinct class of materials, referred to as “Dirac materials”[8], such as graphene [9], topological insulators [10], Dirac [11], andWeyl semimetals [12], is characterized by low-energy fermionicexcitations that behave as massless Dirac particles with lineardispersion. The combination of the intriguing properties of the HMs andthe Dirac materials could give rise to yet another exotic state ofmatter, the so-called Dirac half-metal (DHM), characterized by a bandstructure with a gap in one channel but a Dirac cone in the other[13,14]. Furthermore, if the DHM possesses strong spin-orbit coupling(SOC), it can trigger a gap opening in one spin channel and drive inturn the system in the quantum anomalous Hall effect (QAH) state. Recentfirst-principles electronic structure calculations predicted that thebulk crystal structure of MnF₃, in the hexagonal R3c space group (No.167), is a DHM [15]. Nevertheless, the multiple Dirac cones do notpersist in ultrathin (≈1 nm) MnF₃ films irrespective of the surfaceorientation [16].

While there has been tremendous progress in the field of 2D materialsdisplaying a broad range of electronic and optical properties [9,17,18],most of them in the pristine form are nonmagnetic, thus limiting theirapplications in spintronics. Although magnetism can be introduced bydopants and defects [13,19], a long-range magnetic order has rarely beenobserved experimentally in 2D materials. Interestingly, during the pastyear two teams have observed clear signatures of magnetism in 2D CrGeTe₃[20] and CrI₃ [21] van der Waals materials down to the monolayer limit.However, both these 2D materials are ferromagnetic insulators with lowCurie temperatures of 45 K and 90 K, respectively. Yet, no 2D DHMpristine material has been experimentally synthesized. Consequently,there is an intense current effort on identifying 2D materials torealize such an exotic state that will also satisfy the above materialrequirements.

In order to advance the science in this area, novel 2D materials shouldbe developed that have a wide range of intriguing properties, which makethem highly promising candidates for the next-generation of ultra-lowpower, scalable, non-volatile spintronic devices, including: (1) theyare robust intrinsic half metals ferromagnets with 100% spinpolarization where the band gap of the minority spin channel is large(about 4-5 eV); (2) the majority spin-channel should exhibit a Diraclinear band dispersion leading to the first family of 2D intrinsicferromagnets, which combine two important properties of both halfmetallic behavior and massless Dirac electrons; (3) the Dirac electronbehavior should be robust upon inclusion of spin-orbit interaction; (4)they should exhibit high Fermi velocities up to 3.83×10⁵ m/s comparableto those in graphene; however, unlike graphene, the electrons in MnX₃are fully spin-polarized; (5) the calculated cohesive energies, phonondispersion, and finite-temperature Born-Oppenheimer molecular dynamicssimulations should demonstrate the stability of these compounds andhence their experimental feasibility; (6) they should have large spinmoments (about 4μ_(B) per Mn atom), large exchange interactions andhence high Curie temperatures (higher than 560 K); (7) the materialsshould exhibit giant magnetocrystalline anisotropy energy (MCA) (8.71and 11.86 erg/cm²) with in-plane magnetization orientation.

SUMMARY OF THE SUBJECT MATTER

Ferromagnetic materials are disclosed that comprise at least one Dirachalf metal material. In addition, Dirac half metal materials aredisclosed, wherein the material comprises a plurality of massless Diracelectrons.

In addition, ferromagnetic materials are disclosed that includes atleast one Dirac half metal material, wherein the material comprises aplurality of massless Dirac electrons, wherein the material exhibits100% spin polarization, and wherein the plurality of electrons exhibitultrahigh mobility.

Spintronic devices and heterostructures are also disclosed that includea Dirac half metal material.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1: (a) Top and side views of the 2D MnX₃ crystal structure with theunit cell, where the yellow and blue spheres denote the Mn and X atoms,respectively. The MnX₆ octahedron with each Mn³⁺ ion coordinated to sixX⁻ ions and the hexagonal Brillouin zone (BZ). (b) Top view of variousAFM spin configurations: AFM-Néel (AFM-N), AFM-stripy (AFM-ST),AFM-zigzag (AFM-ZZ), and mixed AFM-N-ST, where the red (black) circlesdenote the up (down) spins.

FIG. 2: (a) Calculated phonon dispersion curves of the 2D MnX₃. (b) Topand side views of atomic structure snapshots from BOMD simulations at1200 K or 900 K.

FIG. 3: (a) and (c) Minority- and majority-spin band structure of the 2DMnF₃ and MnI₃ FM phase using the DFT+U and HSE06 method. (b) and (d) 3Dband structure of the Dirac cone at the K symmetry point and thecorresponding projection on the BZ around the Fermi energy (set atzero).

FIG. 4: Energy and k contribution of atom-resolved (left panel),halogen-p-resolved (middle panel), and Mn-d-resolved (right panel) tothe majority-spin bands for (a) MnF₃ and (b) MnI₃ using the HSE06functional. The color intensity denotes the amplitude of the atom-and/or orbital-resolved character.

FIG. 5: (a) Band structure of the zigzag (left) and the armchair (right)edges of the MnBr₃ ribbon, with the edge states connecting the 2Dvalence and conduction bands. (b) Temperature variation of the Mn³⁺magnetic moment for the 2D MnX₃. (Inset) Temperature variation of theheat capacity.

FIG. 6: (a) Band structures of the 2D MnX₃ determined by the HSE06functional in the presence of SOC, where the Fermi level is set at zero;(b) Electron localization function (ELF) of the side view atomicconfiguration shown on the panel below where the red (blue) color denoteregions where ELF=1 (0) corresponding to accumulation (depletion) ofelectron charge density.

DETAILED DESCRIPTION

As introduced in the Background section, spintronics is a continuouslyexpanding area of research and development merging the areas ofmagnetism and electronics. It exploits the intrinsic electron spindegrees of freedom, in addition to its charge, to create newfunctionalities and new devices. [1] The use of spin offers thepotential advantages of non-volatility, lower power consumption,enhanced data processing speed, and increased integration densitiescompared with conventional semiconductor devices. As discussed above,one important challenge is to discover ferromagnetic thin films whichexhibit high spin polarization which will in turn enhance both thetunnel magnetoresistance (TMR) and spin transfer torque (STT) forread/write operations. Bulk half-metals, with large intrinsic large spinpolarization, are ideal magnetic electrode materials to integrate inspintronics device applications, including Magnetic tunnel junctions(MTJs), [37] giant magnetoresistance devices (GMRs), spin transfertorque, etc. However, the spin polarization of half metals disappearswhen the film thickness is reduced. Thus, it is critical to discovernovel ultrathin ferromagnetic materials, which preserve their halfmetallicity, have ultrahigh electron mobility, and large magneticanisotropy.

Spintronics is a field of nanoscale electronics using the injection,manipulation and detection of the spin of the electron, in addition toits charge, for memory and logic applications. Due to advancements inspintronics, information storage has experienced tremendous growth inthe past decade because it offers opportunities for a new generation ofultralow power, ultrafast, scalable, and nonvolatile devices. MTJs,consisting of two thin ferromagnetic (FM) films separated by a thininsulating oxide layer, are prototypical spintronic devices, where thestate (0 or 1) of the magnetic random access memory (MRAM) bit is storedin the relative orientation (parallel or antiparallel) of themagnetization of the two FM films with different TMR values. [38]However, achieving functional spintronic devices requires thedevelopment of novel magnetic materials with desirable properties andintegration of such diverse materials with atomic-level control. Crucialproperties of the FM thin films include: (1) high spin polarization forlarge TMR, (2) large magnetic anisotropy to ensure room-temperature bitstability, (3) large voltage control of magnetic anisotropy (VCMA) toreduce the magnetic bit switching energy and write voltage, (4) highelectron mobility, and (5) high Curie temperature.

Half-metals (HMs) are a class of bulk materials that are metallic onlyfor one spin direction and semiconducting or insulating for the otherspin direction. [39] Consequently, the spin polarization of theconduction electrons should be 100%, thus providing fully spin-polarizedcurrents. However, one of the current bottlenecks using thin halfmetallic films for MRAM applications, is that they do not retain theirhalf metallic nature and hence the high spin polarization is reduced dueto the confinement of the electrons normal to the surfaces/interfaces.Thus, it is critical to discover novel thin film (two-dimensional)ferromagnetic materials, which are half metallic to ensure huge TMRvalues.

Another distinct class of materials, referred to as “Dirac materials”[8], such as graphene [9], topological insulators [10], Dirac [11] andWeyl semimetals [12], is characterized by low-energy fermionicexcitations that behave as massless Dirac particles with lineardispersion. The combination of the intriguing properties of the HMs andthe Dirac materials could give rise to yet another exotic state ofmatter, the so-called Dirac half-metal (DHM), characterized by a bandstructure with a gap in one channel but a Dirac cone in the other.[13,14] Furthermore, if the DHM possesses strong spin-orbit coupling(SOC), it can trigger a gap opening in one spin channel and drive inturn the system in the quantum anomalous Hall effect (QAH) state. Recentfirst-principles electronic structure calculations predicted that thebulk crystal structure of MnF₃, in the hexagonal R3c space group (No.167), is a DHM [15]. Nevertheless, the multiple Dirac cones do notpersist in ultrathin (≈1 nm) MnF₃ films irrespective of the surfaceorientation [16].

While there has been tremendous progress in the field of 2D materialsdisplaying a broad range of electronic and optical properties [9, 17,18], most of them in the pristine form are nonmagnetic, thus limitingtheir applications in spintronics. Although magnetism can be introducedby dopants and defects [14, 19], a long-range magnetic order has rarelybeen observed experimentally in 2D materials. Interestingly, during thepast year two teams have observed clear signatures of magnetism in 2DCrGeTe₃ [20] and CrI₃ [21] van der Waals materials down to the monolayerlimit. However, these 2D materials are ferromagnetic insulators with lowCurie temperatures of 45K and 90K, respectively. Yet, no 2D DHM pristinematerial has been experimentally synthesized. Consequently, there is anintense current effort on identifying 2D materials to realize such anexotic state that will also satisfy the above material requirements.

In addition, it is equally important to search for 2D half metallicmaterials where the massless Dirac electrons, which exhibit a linearenergy-momentum dispersion (similar to graphene), exhibit ultrahighmobility critical for the operational speed of nanodevices. To date, nosuch 2D Dirac half-metal ferromagnetic materials have been discovered.The emergence of both half metallic magnetic behavior and massless Diracelectrons, if possible, could open up numerous opportunities for 2Dmagnetic and magnetoelectric applications.

Atomically thin layered van der Waals (vdW) crystals provide an idealplatform of two-dimensional (2D) material systems which exhibit a widerange of intriguing properties for emerging functional devices, such asultrafast photodetectors, broadband optical modulators and excitonicsemiconductor lasers.[5] Several experiments have shown that one caninduce extrinsic magnetism in 2D materials through (1) defectengineering; (2) introducing magnetic species; and (3) the magneticproximity effect, whereby 2D materials are placed in contact with othermagnetic substrates. Nevertheless, no 2D crystal with intrinsicmagnetism has yet been discovered. Recently, Huang et. al. demonstratedthe 2D ferromagnetism in exfoliated monolayer chromium tri-iodide (CrI₃)with out-of-plane spin orientation and a Curie temperature of 45 kelvin.However, CrI₃ is not half metal and hence does not exhibit high spinpolarization.

In order to advance the science in this area, a novel and contemplatedfamily of 2D materials has been discovered and is disclosed herein,including MnX₃ (X=F, Cl, Br, and I) that have a wide range of intriguingproperties and contemplated embodiments, which make them highlypromising candidates for the next-generation of ultra-low power,scalable, non-volatile spintronic devices: (1) they are robust intrinsichalf metals ferromagnets with 100% spin polarization where the band gapof the minority spin channel is large (about 4-5 eV); (2) the majorityspin-channel exhibits a Dirac linear band dispersion leading to thefirst family of 2D intrinsic ferromagnets which combine two importantproperties of both half metallic behavior and massless Dirac electrons;(3) the Dirac electron behavior is robust upon inclusion of spin-orbitinteraction; (4) they exhibit high Fermi velocities up to 3.83×10⁵ m/scomparable to those in graphene; however, unlike graphene, the electronsin MnX₃ are fully spin-polarized; (5) the calculated cohesive energies,phonon dispersion, and finite-temperature Born-Oppenheimer moleculardynamics simulations demonstrate the stability of these compounds andhence their experimental feasibility; (6) they have large spin moments(about 4μ_(B) per Mn atom), large exchange interactions and hence highCurie temperatures (higher than 560 K); (7) The MnBr₃ and MnI₃ exhibitgiant magnetocrystalline anisotropy energy (MCA) (8.71 and 11.86erg/cm²) with in-plane magnetization orientation. These values arehigher by a factor of about 5 compared to ferromagnetic materials(CoFeB) currently used in MRAM applications.

Ferromagnetic materials are disclosed that comprise at least one Dirachalf metal material. Contemplated ferromagnetic materials are ultrathin.And as used herein, the term “ultrathin” means it is about or less thanabout 1 nanometer in average thickness. In some embodiments, thematerial is doped with at least one other element. As understood, anysuitable dopant element is contemplated for these materials. Dopants maybe utilized herein for the purpose of modulating or influencing theelectrical, optical, or structural properties of the material ormaterials.

It is understood that contemplated materials have a surface. In someembodiments, the surface is modified with at least one small molecule,at least one defect, at least one additional layer of material, or atleast one element.

In some embodiments, contemplated ferromagnetic materials comprise atleast one monolayer. Contemplated monolayers or layers may all compriseat least one Dirac half metal material, or some of the contemplatedlayers may comprise other materials. Contemplated materials areferromagnetic and monolayer. However, bilayer or thicker layers are alsocontemplated.

In addition, Dirac half metal materials are disclosed, wherein thematerial comprises a plurality of massless Dirac electrons. In someembodiments, a contemplated plurality of electrons exhibits ultrahighmobility. As used herein, the term “ultrahigh” with respect to mobilityis calculated using the Fermi velocity. In order to determine if themobility is ultrahigh, it is compared with the mobility of graphene. Insome embodiments, a contemplated Dirac half metal material exhibits 100%spin polarization.

Contemplated Dirac half metal materials comprise at least one manganesetrihalide. In some embodiments, a contemplated halide comprisesfluorine, chlorine, bromine, or iodine.

In addition, ferromagnetic materials are disclosed that includes atleast one Dirac half metal material, wherein the material comprises aplurality of massless Dirac electrons, wherein the material exhibits100% spin polarization, and wherein the plurality of electrons exhibitultrahigh mobility, which means that one spin channel is insulating,another spin channel is metallic with Dirac. There are no interactions(noise) between different spin channels leading to more efficienttransport, low energy-consumption.

Spintronic devices are also disclosed that include a Dirac half metalmaterial. Contemplated spintronic devices comprise at least one layer ofa Dirac half metal material. In contemplated spintronic devices, theDirac half metal material is two dimensional. Contemplated spintronicdevices may also comprise the ferromagnetic materials disclosed herein.

Heterostructures are also contemplated herein that comprise contemplatedferromagnetic materials. In general, heterostructures are structures orassemblies that comprise more than one different material.

As part of this work, two great challenges in spintronics have beensolved on how to further enhance the performance of ferromagnetic layermaterials with high spin polarization ratio; large magnetocrystallineanisotropy (MCA), excellent carrier mobility, low energy-consumption,and high Curie temperature. The 2D MnX₃ layer materials we discoveredresolve the shortcomings of bulk half-metallic materials in which thehalf-metallic features are lost on going to ultrathin layers and provideopportunities for engineering new magneto-optoelectronic devices withsuperior performance.

In summary, contemplated two-dimensional ‘Dirac half metals’ withferromagnetic ground states and high Curie temperature with intriguingproperties, discussed in detail above and considered contemplatedembodiments, have been discovered that makes them promising for thenext-generation spintronics devices. Research reveals that the proposedMnX₃ layer materials can maintain their ferromagnetism andhalf-metallicity up to 560 K. In addition, this family of materialsexhibits excellent stability. The intrinsic 2D monolayer structures canbe adopted directly as thin film to assemble spintronic devices.

EXAMPLES

Density functional theory (DFT) calculations were carried out using theVienna ab initio simulation package (VASP) [22,23]. The pseudopotentialand wave functions are treated within the projector-augmented wave (PAW)method [24]. Structural relaxations were carried out using thegeneralized gradient approximation as parametrized by Perdew et al.[25]. The plane-wave cutoff energy was set to 500 eV and a 9×9×1 k meshwas used in the Brillouin zone (BZ) sampling for the relaxationcalculations. The band structure was calculated using (i) the DFT+Uapproach [26] (U=3.9 eV) to treat the strong correlations of the Mn delectrons and (ii) the more accurate Heyd-Scuseria-Ernzerhof (HSE06)[27] functional. For the MCA calculations, the SOC was included with a31×31×1 k-point mesh. For the phonon calculations the VASP and PHONOPY[28] codes were employed with a 3×3×1 72-atom supercell to determine thedynamical matrix.

Equilibrium Structural and Magnetic Properties.

The crystal structure of the MnX₃ monolayer, shown in FIG. 1(a),consists of a plane of Mn atoms forming a honeycomb lattice andsandwiched between two X atomic planes, with two Mn and six X atoms per(1×1) unit cell, similar to that of CrI_(3 [21)]. The Mn ions aresurrounded by six first-nearest-neighbor halogens arranged in an edgesharing distorted octahedra, shown in FIG. 1(a). In sharp contrast tothe crystal structure in Ref. [15] where two Mn ions are bonded by asingle anion, two Mn³⁺ in FIG. 1(a) are bonded by two anions.Consequently, the crystal structure in FIG. 1(a) is dramaticallydistinct than any of the surface orientations of bulk MnF₃.

TABLE 1 Calculated equilibrium lattice constant, bond lengths of Mn—Xand Mn—Mn, angle of the X—Mn—X bond, cohesive energy, magnetic moment ofMn³⁺, and MCA per unit area, respectively, for the FM ground state. Wealso list values of the energy difference between the FM ground stateand the AFM-ZZ and AFM-N-ST states, respectively, of the 2 × 2 × 1 unitcell). MnF₃ MnCl₃ MnBr₃ MnI₃ a (Å) 5.36 6.21 6.58 7.08 d_(Mn—X) (Å) 1.962.38 2.55 2.77 d_(Mn—Mn) (Å) 3.09 3.58 3.83 4.08 <Mn_X_Mn (°) 104.2997.44 96.10 94.81 E_(coh) (eV/atom) −4.33 −3.10 −2.70 −2.31 μ (μ_(B))3.92 4.08 4.18 4.27 E_(AFM-ZZ) − E_(FM) (meV) 273 212 248 250E_(AFM-N-ST) − E_(FM) (meV) 115 269 430 436 MCA (erg/cm²) −0.013 −0.46−8.71 −11.86

We have carried total-energy spin-polarized calculations of the 2×2×1unit cell of the ferromagnetic (FM) and various antiferromagnetic (AFM)phases, such as the AFM-Néel (AFM-N), the AFM-zigzag (AFM-ZZ), theAFM-stripy (AFM-SR), and the mixed AFM-N-ST, respectively, shown in FIG.1(b). We find (Table I) that the optimized FM is the ground state forall MnX₃ and that the next highest-energy configuration is the AFM-ZZfor X=Cl, Br, and I and the AFM-N-ST for X=F. Table I lists values ofthe equilibrium lattice constants, the Mn—Mn and Mn—X bond lengths, theX—Mn—X angle, the Mn magnetic moments, and cohesive energies for the FMphase. We also list values of the energy differences, between the mostlikely AFM-ZZ and AFM-N-S and the FM ground state. As expected, thelattice constant, the Mn—Mn and Mn—X bond lengths increase as thehalogen anion's ionic radius increases. For the MnF₃ ML, the Mn—F bondlength of 1.96 Å is close to the value of 1.93 Å in bulk β-MnF₄,implying strong chemical bonding. Similarly, the cohesive energydecreases with an increasing atomic number of the halogen due to thedecreasing electronegativity of the halogen anion.

The magnetic moment per Mn atom of the FM phase, also listed in Table I,increases from 3.92μ_(B) in MnF₃ to 4.27μ_(B) in MnI₃. The magneticmoment is consistent with the +3-oxidation state of Mn and hence the4s⁰3d⁴ electronic configuration. In the octahedral environment of thesix halogens the 3d energy split into a higher-energy e_(g) doublet anda lower-energy t_(2g) triplet, resulting in a spin s=2t³ _(2g)e¹ _(g)electronic configuration for the Mn⁺³ ion according to the Hund's rulecoupling. This is similar to the 2D organometallic honeycomb frameworkwith different embedded transition metals. The electron localizationfunction for MnF₃ displayed in FIG. 4S(b) (Supplemental Material [29]),shows strong localization of the electron density around the metalcations and halogen anions representative of Mn—X ionic bonding.

The MCA per unit area A is, MCA=[E_([100])−E_([001])]/A, where E_([100])and E_([001]) are the total energies with magnetization along the [100]and [001] directions, respectively. The values of MCA, listed in TableI, show that the MCA<0 indicating the in-plane magnetization orientationin all MnX₃'s. Furthermore, the IMCAI increases with increasing halogensize. The value of −0.46 erg/cm² in MnCl₃ is comparable to that of −0.56erg/cm² in the ultrathin Au/FeCo/MgO heterostructure [30] for MRAMapplications. More importantly, the MCA values of 8.71 erg/cm² and 11.9erg/cm² in MnBr₃ and MnI₃, respectively, are about an order of magnitudehigher than that of 1.4 erg/cm² in Ta/FeCo/MgO MRAM nanojunctions. Thegiant MCA values presumably arise from the strong SOC of the heavierX=Br, I, suggesting that introduction of heavy elements in transitionmetal-based films, may be an efficient strategy in enhancing the MCA.

Dynamical and Thermal Stability.

In order to corroborate the dynamical stability of the FM ground stateof the 2D MnX₃ we have carried out both phonon calculations and abinitio Born Oppenheimer molecular dynamics (BOMD) simulations. Thephonon dispersions of the MnX₃, shown in FIG. 2(a), exhibit similaroverall shape with the phonon frequencies softening with increasing massof X. The absence of imaginary frequencies confirms the dynamicalstability of all MnX₃ monolayers. FIG. 2(b) and Supplemental Material,FIG. 1S [29], show the snapshots of the MnX₃ atomic configurations afterannealing for 10 ps at different temperatures of 300, 600, 900, and 1200K, respectively. One can clearly see that the 2D monolayers retain theirhoneycomb atomic structures up to 1200 K for MnF₃, MnCl₃, and MnBr₃, andup to 600 K for MnBr₃ [see FIG. 1S(d) in Supplemental Material [29]].This is consistent with the higher cohesive energies of the lighter MnX₃systems. These results demonstrate that the 2D manganese trihalides areboth dynamically and thermally stable for various room-temperaturespintronic applications.

Electronic Structure.

FIGS. 3(a) and 3(c) show the minority- and majority-spin band structuresof the FM phase for the MnF₃ and MnI₃ monolayer, respectively, employingthe PBE+U (orange curves) and the more accurate hybrid HSE06 (bluecurves) functional. Similar band structures for the MnCl₃ and MnBr₃ aredisplayed in FIGS. 2S(a) and 2S(c) in Supplemental Material [29]. Thesecalculations reveal that all MnX₃ exhibit two fascinating propertieswhich are independent of the exchange correlation functional: (1) Theminority-spin channel is an insulator with an unusually large gap, and(2) the majority-spin channel exhibits Dirac cones at the threehigh-symmetry K points at the Fermi level (E_(F)) for X=F, Cl, Br. Whilethe Dirac cone is in the vicinity of E_(F) for the heaviest MnI₃ at thePBE+U level, the HSE06 functional shifts it at the E_(F). Consequently,we predict that the 2D MnX₃ are intrinsic DHMs. The PBE+U values of theband gap of the minority spin channel are 6.3 eV, 4.33 eV, 3.85 eV, and3.10 eV for the MnF₃, MnCl₃, MnBr₃, and MnI₃, respectively. The PBEfunctional underestimates the gap by about 20% compared to thecorresponding HSE06 values of 7.94 eV, 5.42 eV, 4.79 eV, and 3.89 eV,respectively. The HSE06 values of the Fermi velocities of the Diracelectrons are 3.83×10⁵, 2.40×10⁵, and 2.31×10⁵ m/s for MnF₃, MnCl₃, andMnBr₃, respectively, close to the value of 8×10⁵ m/s in graphene. ForMnI₃ the Fermi velocities of the Dirac electrons and holes are 1.56×10⁵m/s and 3.36×10⁵ m/s, respectively. The combination of 100% spinpolarization and massless Dirac fermions renders this family a naturalcandidate for future applications in spintronics and optoelectronics.

The three-dimensional (3D) majority-spin band profiles around E_(F) nearK are shown in FIGS. 3(b) and 3(d) for MnF₃ and MnI₃, and in FIGS. 2S(b)and 2S(d) in Supplemental Material [29] for MnCl₃ and MnBr₃,respectively. With an increasing atomic number of the halogen, the Diraccone of the conduction band minimum becomes flatter while there is nosignificant change of the cone of the valence band maximum. Thecorresponding projection of the Dirac cones on the 2D BZ around theFermi level are shown in FIGS. 3(b) and 3(d).

The atom- and orbital-resolved majority-spin band structures obtainedwith the HSE06 functional are shown in FIGS. 4(a) and 4(b) for X=F, I,and in FIGS. 3S(a) and 3S(b) in Supplemental Material [29] for X=Cl, Br,respectively. Overall, we find that the linearly dispersivemajority-spin electronic bands at the Fermi energy arise fromhybridization primary of the Mn-derived d_(xz) and d_(yz) (and to asmaller extent of d_(x2-y2)) states with the halogen-derived in-planep_(x) and p_(y) states. The relative strength of the X-p_(x,y) to theMn-d_(xz,yz) contribution increases as the halogen atomic size increasesdown the group, where the Dirac cone is mainly composed of Mn-d derivedstates in MnF₃ and of X-p states in MnI₃. The valence and conductionbands of the Dirac cone are composed mainly of anion-p_(x) and -p_(y)derived states, respectively. The low-energy Dirac cone Mn-d-derivedstates in MnF₃ are in sharp contrast to the corresponding p-derivedstates in graphene. In addition, the weak hybridization of the in-planehalogen-p_(x,y) Dirac states with the underlying substrates willpresumably preserve the Dirac cone, as opposed to the strong out-ofplane π(p_(z)) hybridization of graphene bonds with substrates.

Effect of Spin-Orbit Coupling.

FIG. 6 shows the band structure of MnX₃ determined by the HSE06functional in the presence of SOC. The inclusion of SOC triggers a smallgap opening of −3-10 meV at the high symmetry K point indicating thatthe 100% spin polarization will be maintained at room temperature.Remarkably, the linear band dispersion of the majority spin channel isalso preserved in the presence of SOC for the lighter halogens (X=F, Cl,and Br). On the other hand, the large SOC of the iodine induces (i) agap opening of −91 meV at K, and (ii) splits the doubly degenerateI-derived p_(x,y) states at ˜0.15 eV below E_(F) at r [see FIG. 4(b)]into one band ˜0.1 eV above E_(F) and the other ˜0.3 eV below E_(F),thus preserving the half-metallicity of MnI₃.

To identify the topological properties of the gapped state of thelighter MnX₃, we have calculated the Chern number of MnBr₃, whichexhibits the largest gap. We have employed the Wannier charge centersapproach [31], where the maximally localized Wannier functions wereconstructed from the first-principles calculations including SOC usingthe Wannier90 package [32]. The Chern number for the MnBr₃ monolayer iscalculated from the evolution of the hybrid Wannier charge centers(HWCCs) during a time-reversal pumping process [33,34]. The Chernnumber,

${C - {\frac{1}{ea}\lbrack {{P_{e}^{h}( {2\pi} )} - {P_{e}^{h}(0)}} \rbrack}},$where a is the lattice constant, P_(c) ^(k)(k_(y))=eΣ_(n)(x_(n) k_(y))is the hybrid electronic polarization, and the HWCC, x_(n) k_(y), is asmooth function of k_(y) for k_(y)∈[0, 2π]. Namely, the Chern number canbe viewed as the number of electronic charges pumped across one unitcell in the course of a cycle [35]. We find an odd Chern number, C=−1,for MnBr₃, indicating that it is a QAH insulator with a topologicalnontrivial gap. To corroborate this result we have calculated the bandstructures of the zigzag and armchair edges of the MnBr₃ ribbon, whichare displayed in FIG. 5(a). The emergence of a single chiraltopologically protected gapless edge state near E_(F) connecting the 2Dvalence and conduction bands is consistent with the calculated Chernnumber. Thus, we predict that MnF₃, MnCl₃, and MnBr₃ provide a promisingplatform for exploring the QAH effect at ˜60-100K, which is three ordersof magnitude higher than the temperature of 100 mK at which the QAH wasrecently observed in Cr-doped Bi₂Se₃ films [36].Magnetic Properties.

Since the values of the X—Mn—X angle, listed in Table I, are close to90° the superexchange interaction between two nearest-neighbor (NN) Mnatoms mediated by X are expected to be FM and dominant (the direct AFMexchange interactions are weak due to the large Mn—Mn distance). Theexchange interaction parameters J_(i,j) are determined by expressing theDFT total energies of the various FM and AFM configurations to theHeisenberg spin Hamiltonian,

${\hat{H} = {{- J_{i,j}^{\prime}}{\sum\limits_{i,j}{{\hat{S}}_{i}{\hat{S}}_{j}}}}},$detined on a honeycomb lattice, where |S=2|. The second- and third-NNexchange interactions are smaller than the first-NN interaction J₁,which increases from 3.8 meV in MnF₃ to 10 meV in MnI₃. Using theDFT-derived exchange interactions, we have determined the Curietemperature Tc, using Metropolis Monte Carlo simulations of the 100×1002D honeycomb lattice with periodic boundary conditions. The temperaturevariation of the Mn³⁺ magnetic moment for the MnX₃ is shown in FIG.5(b). We find that Tc increases from 450K in MnF₃ to 720K in MnI₃, whichare higher than that of in CrI₃. The inset shows the temperaturevariation of the specific heat which peaks at Tc.

In summary, contemplated embodiments show that the 2D pristine manganesetrihalides is a family of intrinsic Dirac half-metals which exhibitsmany unique properties, including 100% spin polarization, massless Diracfermions with high carrier mobility, large magnetic moments, high Curietemperatures, and large in-plane magnetic anisotropy. Consequently, theymeet many requirements of high-efficiency spintronic applications. Wedemonstrate that the MnX₃ are dynamically and thermodynamically stableup to high temperatures and hence could be synthesized experimentally byan exfoliation process commonly employed in other 2D van der Waalcrystals. Lastly, in contrast with conventional FM films, which interactstrongly with the underlying substrates, the electronic and magneticproperties of these 2D van der Waals DHMs can remain on substrates.

Thus, specific embodiments, methods of use, and production oftwo-dimensional Dirac half-metal materials and ferromagnets forspintronic devices have been disclosed. It should be apparent, however,to those skilled in the art that many more modifications besides thosealready described are possible without departing from the inventiveconcepts herein. The inventive subject matter, therefore, is not to berestricted except in the spirit of the disclosure herein. Moreover, ininterpreting the specification, all terms should be interpreted in thebroadest possible manner consistent with the context. In particular, theterms “comprises” and “comprising” should be interpreted as referring toelements, components, or steps in a non-exclusive manner, indicatingthat the referenced elements, components, or steps may be present, orutilized, or combined with other elements, components, or steps that arenot expressly referenced.

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The references included below are incorporated by reference herein.

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The invention claimed is:
 1. A ferromagnetic material, comprising atleast one Dirac half metal material that comprises two-dimensionalmanganese trihalide crystals that are ferromagnetic, wherein thematerial exhibits 100% spin polarization, a high Curie temperature of560 K or higher, and a band gap of the minority spin channel of at leastabout 4 eV.
 2. The ferromagnetic material of claim 1, wherein theferromagnetic material is ultrathin.
 3. The ferromagnetic material ofclaim 2, wherein ultrathin is about 1 nanometer in thickness.
 4. Aspintronic device comprising the ferromagnetic material of claim
 2. 5.The ferromagnetic material of claim 1, wherein the ferromagneticmaterial comprises at least one monolayer.
 6. The Dirac half metalmaterial of claim 1, wherein the halide of the trihalide comprisesfluorine, chlorine, bromine, or iodine.
 7. The ferromagnetic material ofclaim 1, wherein the material is doped with at least one other element.8. The ferromagnetic material of claim 1, wherein the material has asurface, and the surface is modified with at least one molecule, atleast one defect, at least one additional layer of material, or at leastone element.
 9. A heterostructure comprising the ferromagnetic materialof claim
 1. 10. The heterostructure of claim 9, wherein theheterostructure is vertical or lateral.
 11. A spintronic devicecomprising a ferromagnetic Dirac half metal material that comprisestwo-dimensional manganese trihalide crystals that are ferromagnetic,wherein the material exhibits 100% spin polarization, a high Curietemperature of 560 K or higher, and a band gap of the minority spinchannel of at least about 4 eV.
 12. The spintronic device of claim 11,comprising at least one layer of a Dirac half metal material.